# Relativistic doppler effect - inconsistency in my derivations

• osturk
In summary: Looking at your equation, frame 1 should be θ=π/2 and frame 2 should be θ=0. This is the special case of transverse doppler effect (θ=π/2). [..]The relativistic Doppler effect: when a zero frequency shift or a red shift exists for sources approaching the observer.Annalen der Physik (Berlin) 523, No. 3, 239 - 246 (2011); http://arxiv.org/abs/1006.4407

#### osturk

Hello everyone,

I'm trying to calculate the doppler shift in frequency of a moving source. I'm approaching the problem from two different frames of reference and getting inconsistent results. what am I missing here?

consider the special case of transverse doppler effect (θ=π/2). light reaches observer from y-direction.

reference frame 1: source traveling towards x-direction. observer at rest.
Δto=1/fo - observer's time between subsequent wave crests.
due to time dilation, source's time should be running slower Δ$\tau$s=Δto/$\gamma$, so actual frequency of source is 1/Δ$\tau$s=fs=fo*$\gamma$ (the correct relation)

reference frame 2: observer traveling towards x-direction. source at rest.
Δts=1/fs - source's time between subsequent wave crests.
due to time dilation, observer's time should be running slower Δ$\tau$o=Δts/$\gamma$, so frequency observed is 1/Δ$\tau$o=fo=$\gamma$fs
which leaves actual frequenct of source fs=fo/$\gamma$

why are these two results seemingly inconsistent?

You are doing the same thing in both frames but simply renaming the source and the observer which is the correct thing to do and is showing the reciprocal relationship between the source and the observer but you should have left the second equation as fo=fs*γ. The actual frequency of both the source and the observer is the same in their own rest frame and they each see the same Doppler frequency shift of the other one. Isn't that what your equations indicate? And isn't that the whole point of Relativistic Doppler?

osturk said:
Hello everyone,

I'm trying to calculate the doppler shift in frequency of a moving source. I'm approaching the problem from two different frames of reference and getting inconsistent results. what am I missing here?

consider the special case of transverse doppler effect (θ=π/2). light reaches observer from y-direction.

reference frame 1: source traveling towards x-direction. observer at rest.
Δto=1/fo - observer's time between subsequent wave crests.
due to time dilation, source's time should be running slower Δ$\tau$s=Δto/$\gamma$, so actual frequency of source is 1/Δ$\tau$s=fs=fo*$\gamma$ (the correct relation)

reference frame 2: observer traveling towards x-direction. source at rest.
Δts=1/fs - source's time between subsequent wave crests.
due to time dilation, observer's time should be running slower Δ$\tau$o=Δts/$\gamma$, so frequency observed is 1/Δ$\tau$o=fo=$\gamma$fs
which leaves actual frequenct of source fs=fo/$\gamma$

why are these two results seemingly inconsistent?

Here is a paper which might be helpful to you:

The relativistic Doppler effect: when a zero frequency shift or a red shift exists for sources approaching the observer
Annalen der Physik (Berlin) 523, No. 3, 239 - 246 (2011); http://arxiv.org/abs/1006.4407

Do you know, a light source, when it is approaching (moving closer to) the observer, may cause a red shift?

osturk said:
Hello everyone,

I'm trying to calculate the doppler shift in frequency of a moving source. I'm approaching the problem from two different frames of reference and getting inconsistent results. what am I missing here?

consider the special case of transverse doppler effect (θ=π/2). light reaches observer from y-direction. [..]

Hi, what you missed is rather basic: the angle of light propagation wrt frame 2 is not equal to that wrt frame 1. Thus you must:
1. specify in which frame θ = 90 degrees
2. calculate the angle in the other frame, and account for the corresponding Doppler effect.

## 1. What is the relativistic doppler effect?

The relativistic doppler effect is a phenomenon in physics that describes the change in frequency of a wave (such as light or sound) due to the relative motion between the source of the wave and an observer.

## 2. Why is there an inconsistency in the derivations of the relativistic doppler effect?

The inconsistency in derivations of the relativistic doppler effect arises due to differences in assumptions and approximations made by different scientists. Additionally, the relativistic doppler effect is a complex phenomenon that can be described using different mathematical models, leading to variations in the derivation process.

## 3. How can I understand the relativistic doppler effect better?

To understand the relativistic doppler effect, it is important to have a strong understanding of concepts such as special relativity, wavelength, frequency, and the speed of light. It may also be helpful to study different derivations and models of the effect to gain a more comprehensive understanding.

## 4. What are the practical applications of the relativistic doppler effect?

The relativistic doppler effect has numerous practical applications, including in astronomy, where it is used to determine the motion and distance of celestial objects. It is also utilized in various technologies, such as radar and GPS systems.

## 5. How does the relativistic doppler effect differ from the classical doppler effect?

The classical doppler effect only accounts for the change in frequency of a wave due to the relative motion between the source and observer. The relativistic doppler effect, on the other hand, takes into consideration the effects of special relativity, which includes time dilation and length contraction, on the observed frequency of the wave.

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